There are two differences however. First, GFA restricts the group specific error covariance to isotropic structure. That is, \(\Sigma ^{(g)}=\sigma ^{2}_g\mathbf {I}\)  for \(g=1, \cdots , G\) . Second, in order to model the dependency structure between any subsets of groups and resolve indeterminacy issues, GFA imposes a structural sparsity prior over the columns of the projection matrix \(\mathbf {W}\) . The Automatic Relevance Determination (ARD) prior [1]} over the \(G \times D\)  group specific columns of the projection matrix \(\mathbf {W}\)  can be specified as follows.
\(\begin{split}\mathbf {w}^{(g)}_d&\sim \mathcal {N}\left(0,{(\alpha ^{(g)}_{d})}^{-1}\mathbf {I}\right) \quad g=1, \cdots , G; \quad d=1,\cdots ,D \\log(\alpha ) &= UV^\intercal + \mu _u \textbf {1}^\intercal + \textbf {1}\mu _v^\intercal \\p(U) &= \prod _g^G{\prod _r^R{\mathcal {N}(u_{g,r}|0, (\lambda =0.1)^{-1})}} \\p(V) &= \prod _d^D{\prod _r^R{\mathcal {N}(v_{d,r}|0, (\lambda =0.1)^{-1})}}\end{split}\) 
